## Abstract

Ultrashort-pulse laser surface and bulk nano- and micromachining of dielectrics have multiple promising applications in micro-optics, microfluidics, and memory storage. The fundamental principles relate intrinsic inter-band multi-photon (MPA) and laser-induced intra-band free-carrier absorption (FCA) to particular ablation mechanisms and features. These principles are yet to be quantified into a complete set of basic experimental laser-matter interaction parameters, describing photoexcitation, relaxation, and final ablation. In this study, we considered the characteristic double-crater structure of single-shot ablation spots on dielectric surfaces and single-shot transmission spectra to extract crucial information about the underlying basic processes of ultrafast photoexcitation and laser energy deposition. Specifically, energy-dependent crater profiles and accompanying prompt self-phase modulation (SPM) spectral broadening were studied in single-shot surface ablation experiments on fluorite (CaF_{2}) surface photo-excited by tightly focused 515- or 1030-nm, 300-fs laser pulses. Crater size dependence demonstrated two slopes, scaling proportionally to the squared focal 1/e-radius at higher energies (intensities) for larger ablated spots, and a much smaller squared 1/e-radius at lower energies (intensities) for (sub) micron-wide ablated spots, indicating a transition from 1D to 3D-ablation. As a result, these slopes were related to lower-intensity wavelength-dependent multi-photon inter-band transitions and wavelength-independent higher-intensity linear absorption in the emerging near-critical electron-hole plasma (EHP), respectively. Crater depth dependences on the local laser intensity fitted in the corresponding ranges by multi- and one-photon absorption provided the corresponding absorption coefficients. Spectral broadening measurements indicated even values for the red and blue shoulders of the laser pulse spectrum, representing the SPM effect in the weakly excited fluorite at the leading pulse front and providing the corresponding Kerr coefficient. In the second regime, the blue-shoulder broadening value saturated, indicating the appearance of near-critical plasma screening at the trailing pulse front, which is consistent with our calculations. These complementary experiments and related analysis provided an important set of key basic parameters, characterizing not only surface ablation, but also propagation of high-intensity ultrashort laser pulses in bulk fluorite, and enabling precise forecasting of optimal energy deposition for high-efficiency ultrashort-laser micro-structuring of this dielectric material.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Ultrashort-pulse (mostly, femtosecond/fs) laser ablative nano- and micromachining on dielectric surfaces and in volume materials is employed both in traditional applications, such as waveguide writing [1], fabrication of birefringent elements [2], 3D optical memory based on nano-void arrays [3], as well as innovative emerging fields such as optical vortex generation [4], holographic recording [5], and 5D optical memory storage [6]. These applications differ in the volume density of the deposited laser energy, enabling structural rearrangement [7], periodic nanograting fabrication [8], hollow nanovoid formation [9], or microchannel drilling [10], with varying amount of transfer/removal of the ablative material. Fundamental femtosecond-laser energy deposition mechanisms are well-known [11–19] and are related to multi-photon (MPI) or tunnel inter-band intrinsic photoionization (TI), or avalanche ionization (AI) via inverse bremsstrahlung heating of photo-generated free carriers. However, unambiguous or straightforward quantification of their key parameters, which change both in spatial and temporal domains, directly from experimental optical transmission [20–21], reflection [22], interferometric [23], or holographic [24,25] measurements is yet to be achieved. This hinders predictive linking between the input laser, focusing, material parameters, and the resulting volume energy densities and microstructures.

During femtosecond-laser ablation on dielectric surface crater depth [26–28], ablated volumes per pulse [29,30] and crater diameters [31–33] appear to be the most crucial machining parameters, which affect the efficiency, as well as spatial, longitudinal, and transversal (lateral) resolutions of ablation. Usually, these parameters are analyzed separately, without a comprehensive analysis of crater profiles that involve the diameter of the external ablation rim, neighboring above-threshold shallow crater, and the central deep depressions [34]. Specifically, numerous double-crater structures of single-shot ablated spots have been observed [35–39], with their external shallow, even flat (“gentle” ablation [36]) and central ultra-deep (“strong” ablation [36]) craters). Meanwhile, direct relationships between such important ablation features and specific regimes of photoexcitation/electron dynamics are yet to be established.

Recently, a two-step analysis procedure was proposed [40,41] for single-shot crater profiles on dielectric surfaces. Both radii and depths are simultaneously considered, covering the entire range of surface ablation modifications, from the characteristic nanometer-deep craters to submicron or even multi-micron depths. First, characteristic ranges of non-linear energy deposition were revealed through the Liu analysis [42]. For local ablation related to the energy deposition spot above the ablation threshold, the Gaussian energy distribution over the focal spot with a focal 1/e-radius W_{foc} results in energy deposition, which is proportional to the Nth power of laser intensity, *I*^{N}, providing the squared characteristic ablation radius R_{abl}^{2} = W_{foc}^{2}/N [28,40,41]. This procedure enables the determination of multi(N)-photon absorption processes in qualitative agreement with the conventional femtosecond-laser ionization dynamics [11–23,43–46]. Moreover, at higher femtosecond-laser intensities, linear free-carrier (near-critical electron-ion plasma) absorption could be identified according to R_{abl}^{2} ≈ W_{foc}^{2} [28,40,41]. Second, the identified intensity ranges for different absorption processes were used for non-linear fitting of the corresponding crater depths across their radial profiles as a function of radial laser intensity [28,40,41]. This is a well-defined manner of deriving absorption coefficients [28,40,41] compared to multi-photon absorption (MPA) approximations [26,27].

In this study, we investigated in detail a two-crater structure on fluorite surfaces CaF_{2}(111) photo-excited by tightly focused 515- or 1030-nm, 300-fs laser pulses. Comparing to multi-component silica glasses, this “clean” material was chosen for its crystalline character with the well-defined optical, electronic, lattice, and transport parameters, with its electronic band and absorption spectra [47] presented in Fig. 1. In this material, lack of any considerable density of defect or impurity states in the bandgap ensures no their effect on not only near-IR, visible, or UV photoexcitation, but also carrier trapping, recombination, and transport dynamics. Determining the two-slope dependence of crater size and depth on pulse energy (intensity) enabled the identification of key laser energy deposition processes in the material in the context of 3D-1D ablation transition, while inhomogeneous spectral broadening indicated the plasma dynamics during the pump pulse. Based on these results, we explain the different effective squared beam radii as fluence calibration slopes in their relationship to actual energy deposition mechanisms, and derived their numerical laser absorption and transport parameters, which are crucial for sub-wavelength ultrashort-laser nanomachining.

## 2. Experiment details

Laser ablation and spectral transmittance studies were performed to characterize the high-intensity non-linear interactions between focused femtosecond laser pulses, having a plane phase wave-front and well-defined waist peak intensity, on the CaF_{2} surface. The schematic of the experimental setup is shown in Fig. 2(a).

In these experiments, we used 300-fs full-width at half maximum (FWHM), quasi-monochromatic 515-nm (FWHM Δλ ≈ 1.7 nm) or 1030-nm (FWHM Δλ ≈ 7 nm) laser pulses (TEM_{00}-mode, M^{2} ≤ 1.07, 1/e-radii σ(1030 nm) ≈ 10 mm, and σ(515 nm) ≈ 8 mm). The pulses were tightly focused onto a 2-mm-thick optically-polished UV-grade CaF_{2} slab by a microscope objective with numerical apertures (NA) of 0.25 (effective focal distance *f*_{0.25 }= 16 mm) into the wavelength-dependent focal spots with the 1/e-radii *w*_{foc}(515 nm) ≈ 2 μm and *w*_{foc}(1030 nm) ≈ 4 μm [48] (the calculated 1/e-radii *w*_{0}(515 nm) = λ*f*_{0.25}/(2πσ*) ≈ 1.7 μm and *w*_{0}(1030 nm) = λ*f*_{0.25}/(2πσ*) ≈ 3.5 μm) owing to the limited objective aperture. The corresponding calculated wavelength-dependent Rayleigh lengths in fluorite, which are relevant for our SPM spectral broadening studies as the corresponding effective interaction lengths, were *z*_{CaF2}(515 nm) = λ*f*_{0.25}^{2}/(4πn_{CaF2}σ^{2}) ≈ 12 μm and *z*_{CaF2}(1030 nm) = λ*f*_{0.25}^{2}/(4πn_{CaF2}σ*^{2}) ≈ 24 μm, where n_{CaF2}(500 nm) = 1.44 and n_{CaF2}(1.0 μm) = 1.43 are the corresponding refractive indices [47]. For 0.65-NA focusing at the 1030-nm wavelength (*f*_{0.65} = 4 mm), the employed focusing parameters were *w*_{foc}(1030 nm) ≈ 0.9 μm (the calculated 1/e-radii *w*_{0}(1030 nm) = λ*f*_{0.65}/(2πσ*) ≈ 0.84 μm) due to the limited objective aperture in air and *z*_{CaF2}(1030 nm) = λ*f*_{0.65}^{2}/(4πn_{CaF2}σ*^{2}) ≈ 1.5 μm in fluorite. All these focal parameters are summarized in Table 1.

Single-shot laser exposition occurred at the variable incident pulse energies *E* = 0.6-3.6 μJ (1030 nm) and 0.1-1.3 μJ (515 nm), providing the NA-dependent peak fluence *F*_{0} = *E*/(πw_{foc}^{2}) ≈ 1.2-110 J/cm^{2} (1030 nm) and 0.8-11 J/cm^{2} (515 nm), peak intensity *I*_{0} = *F*_{0}/τ≈ 4-370 TW/cm^{2} (1030 nm) and 3-37 TW/cm^{2} (515 nm).

The CaF_{2} slab was arranged on a computer-driven motorized 3D-positioning platform. It was scanned from shot to shot to produce a linear pattern of ten 120-μm spaced single-shot micro-craters at each pulse energy, which were flat and shallow (≈25-nm deep) near the crater edge and deep in the center (Figs. 2(b) and 2(c)). The crater profiles were characterized by atomic force microscopy (AFM). AFM topography scans were performed over 10 μm × 10 μm CaF_{2} surface spots at 0.04-μm steps through a microscope Solver Pro P6 (NT-MDT) in a semi-contact mode with a 10-μm long high-resolution silicon probe HA_NC (tip curvature – 10 nm, tip angle – 30°) mounted at the angle of 20° in the scan head “Smena”.

Similarly, spectral acquisition of the pump-pulse transmission was performed on fresh CaF_{2} surface spots in a scanning mode over 50 laser pulses at each pulse energy (Fig. 2(d)). A collecting silica-glass optical fiber was arranged just behind the rear sample surface without any additional collecting optics and coupled to a PC-controlled, single-grating spectrometer ASP-150SF.

## 3. Results and discussion

#### 3.1 Crater profiles and their interpretation

The profiles of the two-crater single-shot ablated spots provide two important characteristics of laser-matter interactions, namely energy-dependent radii of the external shallow crater and the central deep crater. Further, we acquired radial depth variations across the focal spots, which can be converted into local intensity- or fluence dependence of depth. Because the rather narrow and deep central craters could be perturbed during material removal by non-local effects such as lateral melt displacements and redeposition, the control curves of the maximal crater depth in the central peak intensity points were used for comparison.

### 3.1.1 a) Analysis of crater diameters

We used the squared radii *R*_{abl}^{2} of the external shallow and central deep craters versus the natural logarithm of incident pulse energy, ln*E*, to characterize the focusing and energy deposition conditions [28,40,41]. According to the Liu analysis [42], the surface modification features appearing above the threshold *I*_{M} for the Gaussian beam with energy *E* and 1/e-radius *w*_{foc},

This expression can be employed for the linear absorption/energy deposition on the surface, directly replicating the focal energy distribution, resulting in

providing*w*

_{foc}

^{2}as the slope of the

*R*

_{M}

^{2}-ln

*E*dependence and the modification threshold energy

*E*

_{M}(fluence

*F*

_{M}, intensity

*I*

_{M}). In a more general case, for example, for (sub)micrometer-wide focal spots on “thermally thin” metallic films, the lateral thermal conductivity expands the energy deposition spot during the hot-carrier relaxation stage [49] and, later, till the picosecond- or multi-picosecond-scale ablation onset (depending on femtosecond-laser ablation mechanisms [50]).

In dielectrics, where multi(N)-photon absorption of optical photons is the key initial process of energy deposition at low and moderate intensities of ultrashort laser pulses, the effective focal spot size ∼ *w*_{foc}/√N is determined by the main energy deposition region

However, for threshold-like modification processes such as ablation, the squared modification radius again reads similarly to Eq. (3).

*N*-independent [51]. This is because the threshold intensity

*I*

_{M}still follows a Gaussian distribution with

*N*= 1 (see Eq. (2)).

Our quantitative examination of the experimental data for both the 1030-nm and 515-nm wavelengths indicates that *R*_{abl}^{2}-ln*E* curves in Fig. 3 exhibit two characteristic ranges: *range #1* for small and *range #2* for large pulse energies, coinciding with small and large ablated spots, respectively. At smaller pulse energies, the derived slopes *w*_{1}^{2}(1030 nm) ≈ 0.40 μm^{2} at NA = 0.65 and *w*_{1}^{2}(515 nm) ≈ 0.87 μm^{2} at NA = 0.25 are unexpectedly almost two-fold and five-fold lower than their corresponding focal 1/e-radii -radius *w*_{foc}(1030 nm)^{2} ≈ 0.9^{2} μm^{2} and *w*_{foc}(515 nm)^{2} ≈ 2^{2} μm^{2} (Fig. 3), respectively. In contrast, at higher pulse energies for large ablated spots, the curve slopes *w*_{2}^{2} (1030 nm) ≈ 0.95 μm^{2} at NA = 0.65 and *w*_{2}^{2}(515 nm) ≈ 4.3 μm^{2} at NA = 0.25 are in good agreement with their corresponding regular squared values *w*_{foc}^{2}.

The observed integer reduction of the curve slopes in the low-energy (low-intensity) range could indicate multi-photon — 2, 3, or 5-photon — absorption in the material at these particular wavelengths, while the linear free-carrier absorption at high energies (intensities) is in agreement with previous interferometric observations [23]. However, in Eqs. (4), and (5) we have revealed that threshold-like multi-photon processes do not change the curve slopes. Hence, the corresponding low-energy crater sizes appear to be enormously large ablation features.

A possible explanation is as follows. For very small ablation spots (in both Figs. 3(a) and 3(b) - *R*_{abl}^{2} <1 μm^{2}), a transition is expected versus the decreasing pulse energy *E* from 1D in-depth ambipolar diffusion of EHP and the following thermal diffusion over the energy deposition distance *L*_{D}, to their 3D hemispherical diffusion within the ultra-narrow, (sub) micron-wide focal and ablation spots. The transition occurs during the characteristic ablation onset time [49], which is a characteristic of broad focal and ablation spots. In a phase explosion regime, implying a hydrodynamic expulsion of supercritical fluid, the characteristic ablation onset time is determined by *Z*/*C _{l}* ∼10-100 ps, where

*Z*is the thickness of the material in the supercritical state (volume energy density ε ≥ ε

_{abl}) and

*C*is the longitudinal sonic velocity. Then, for large focal spots

_{l}*w*

_{foc}»

*L*

_{D}with the effective nonlinear energy deposition/ablation scale

*L*

_{D}«

*R*

_{abl}«

*w*

_{foc}in the axial coordinates,

*R*

_{abl}<

*w*

_{foc}<

*L*

_{D}in radial coordinates, resulting in Eq. (8), which is qualitatively similar to Eq. (5), but with the pre-factor

*L*

_{D}

^{2}, rather than

*w*

_{foc}

^{2}

As a result, the low-energy values *w*_{1}(1030 nm) ≈ 0.63 μm at NA = 0.65 and *w*_{1}(515 nm) ≈ 0.94 μm at NA = 0.25 can be related to the energy deposition distance *L*_{D,S} along the surface during the characteristic ablation onset time, which could be represented as follows [52]:

_{drift}is the EHP drift distance; D

_{eh}and τ

_{eh}are the EHP ambipolar diffusion coefficient and EHP lifetime, respectively. χ

_{T}and τ

_{T}are the thermal diffusivity and heat diffusion time until the ablation onset; the entire expression in brackets representing the squared transport length σ

_{S}

^{2}. Similarly, the bulk deposition length

*L*

_{D,B}is defined as where the new term δ

_{abs}

^{2}corresponds to the squared effective absorption depth, while the term in the brackets introduces the same squared bulk transport length σ

_{B}

^{2}.

At 1030-nm wavelength and NA = 0.65, for *N* = 9 (Fig. 1) the focal contribution to *L*_{D,S} ≈ *w*_{1}≈ 0.63 μm (*L*_{D,S}^{2} = 0.40 ± 0.04 μm^{2}) in the MPA assumption tends to *w*_{foc}/√9 ≈ 0.3 μm, where the rest value = 0.54 ± 0.02 μm is the transport length σ_{S} (1030 nm). At 515-nm wavelength for *N* = 5 (Fig. 1), the focal contribution to *L*_{D,S} ≈ *w*_{1}≈ 0.93 μm (*L*_{D,S}^{2} = 0.87 ± 0.28 μm^{2}) in the MPA assumption tends to *w*_{foc}/√5 ≈ 0.9 μm, where the rest value as the transport length σ_{S}(515 nm) = 0.3 ± 0.3 μm is of the same order as at the 1030-nm laser wavelength. Typically, these lengths could be different at different focusing conditions realizing different EHP and temperature gradients. These transport lengths are expected to determine also the in-depth laser energy deposition through the EHP and thermal diffusion processes, and the final ablation depth, as considered next in crater depth analysis for both small and large craters in fluorite.

### 3.1.2 b) Analysis of crater depths

Purely “optical” analysis of the crater depth dependence on laser intensity in the ranges *I*_{abl,1} ≤ *I*_{0} ≤* I*_{abl,2} and *I*_{0} ≥* I*_{abl,2} could be performed in approximations of pure MPA and free-carrier absorption (FCA) regimes, enabling quantification of their corresponding coefficients. This was done via analysis of the acquired crater profiles, assuming that, across the ablated spot, the ablation threshold is achieved not only on the surface, but also at a certain depth *Z* via the nonlinear transmission of the incident intensity *I* [28,40,41]. The crater depth dependence on local intensity *Z*(*I*) is demonstrated in Fig. 4 for both these ranges, where the depth increases slowly in the MPA regime for lower *I* and much faster in the FCA regime for higher *I*. Following previous studies [28,40,41], in the first range, the crater depth variation versus *I* was fitted considering the *N*-photon absorption as follows:

_{N}is the N-absorption coefficient. Specifically, at 515 nm, this procedure yields a very large effective 5-photon absorption coefficient, β

_{5}≈ 10 cm

^{7}/TW

^{4}or ≈4 × 10

^{3}cm

^{7}/J

^{4}(compared to β

_{5}≈ 16 cm

^{7}/J

^{4}in fused silica [53] and β

_{5}≈ 5 × 10

^{−5}cm

^{7}/TW

^{4}in sapphire [28] at 800 nm), not accounting for the femtosecond-laser reflection at the air/CaF

_{2}interface. At 1030 nm, the same procedure gives an effective 2-photon absorption coefficient, β

_{2}≈ (4 ± 1) cm/GW, again neglecting the femtosecond-laser reflection at the air/CaF

_{2}interface. One potential ambiguity of such MPA fitting could be a possible spallative, fluence-independent character of the shallow crater in the intensity range, which is, however, not distinct in our study and, in general, is not justified for (sub)micrometer sized craters.

In the high-intensity range (*I*_{0} ≥* I*_{abl,2}) the logarithmic approximation of the linear absorption regime,

^{4}cm

^{-1}at 1030 nm (Fig. 4(a)) and (1.8 ± 0.2) × 10

^{4}cm

^{-1}at 515 nm (Fig. 4(b)), potentially indicating the strong FCA process in the near-critical EHP, which is more pronounced for the near-IR laser radiation.

Our analysis according to Eq. (10), including the energy transport depths σ_{B}(1030 nm) = 0.54 ± 0.02 μm and σ_{B}(515 nm) = 0.3 ± 0.3 μm, indicates their magnitudes to be comparable to the abovementioned values 1/α, including the transport length σ_{B}. Moreover, since the maximal (plateau) crater depth at higher intensities approaches rather similar wavelength-independent magnitudes ≈ 350-400 nm in the high-intensity range (Fig. 4), this may indicate the predominant contribution of the EHP/thermal transport in the energy deposition. This implies that the optical absorption depth δ_{abs} « 1/α ∼ 0.1–1 μm^{-1} could be realized as a strong FCA process in near-critical EHP. Despite the much stronger FCA for 1030-nm femtosecond laser pulses, the much higher threshold intensities in Fig. 4 demonstrate the much lower efficiency of overall photoionization for the longer-wavelength photons, which is in good agreement with previous studies [11,12].

Finally, in the low-intensity range, the shallow craters exhibited nearly constant depths Z ≤ 50 nm within the experimental error bars, representing a slightly violent ablation character (see (Figs. 2(b), 2(d), 2(e)). Such a double-crater structure with the external shallow and central ultra-deep crater is rather universal and is well known for many dielectric materials [11,35].

#### 3.2 SPM spectral broadening

Kerr self-focusing coefficients *n*_{2} and cumulative EHP appearance at the trailing pump-pulse part were revealed during the 1030-nm and 515-nm femtosecond-laser pump pulses in our spectral broadening studies (see Figs. 2(f) and 2(g)). The broadening magnitude in the spectral inflection points follows the time-derivative of the pulse envelope, according to the following equation for the spectral component spectral component at a wavelength λ_{0} over the propagation length *L* [54]:

*c*is the velocity of light in vacuum at a wavelength λ

_{0}, and

*g*(

*t*) is the pump-pulse intensity envelope; in our case, L ≈ z

_{CaF2}. Here, the “red” shoulder broadening is related to self-phase modulation (SPM) at the leading pump-pulse edge, while the “blue” one corresponds to its trailing edge [54]. In this study, on the “red” pump-pulse shoulder, a linear intensity-dependent increase of such pure Stokes SPM broadening δλ was observed with slopes

*K*≈ 0.2 (1030 nm) and 1.4 (515 nm) nm × cm

^{2}/TW (Fig. 5). These intensity-independent slopes imply negligible EHP screening at the leading pump-pulse fronts, enabling the evaluation of their corresponding Kerr coefficients

*n*

_{2}(1030 nm) and

*n*

_{2}(515 nm). In weakly excited CaF

_{2}represented in Eq. (13), the wavelength- and NA-dependent focal Rayleigh lengths

*z*

_{CaF2}can be substituted for

*L*(Table 1). As a result, the derived values

*n*

_{2}(1030 nm) = (0.1-0.2) × 10

^{−16}cm

^{2}/W and

*n*

_{2}(515 nm) = (2.5 ± 0.5) × 10

^{−16}cm

^{2}/W are in agreement with the known values

*n*

_{2}(800 nm) ≈ 1 × 10

^{−16}cm

^{2}/W [55],

*n*

_{2}(2-3 μm) ≈ 2 × 10

^{−16}cm

^{2}/W for CaF

_{2}[56,57]. At longer wavelengths a lower near-IR Kerr coefficient value is expected because of its predicted scaling relationship ∝1/λ

^{2}[58].

In the range #1 in Fig. 5, representing lower femtosecond-laser intensities the “red” and “blue” broadening is uniform; however, at higher intensities in the range #2, the “blue” component saturates first and even decreases, while the “red” component persists at much higher intensities. Since the “blue” shoulder broadens at the trailing pulse front, its saturation and drop could indicate the screening effect of the accumulated opaque near-critical EHP, emerging in CaF_{2} at the trailing edge of the transmitted femtosecond-laser pulses and decreasing their laser intensity. The same saturation and drop for the “red” shoulder exhibits dense EHP formation already at the leading 1030-nm pulse front, screening the four-wave interactions, related to the Kerr effect, with much stronger screening for the “blue” shoulder (Fig. 5(a)).

#### 3.3 Plasma parameters

Numerical analysis was performed to relate the measured effective linear FCA coefficient α ≈ (1.8 ± 0.4) × 10^{4} cm^{-1} to the peak EHP density during the pulse. Prompt optical constants of the photo-excited fluorite versus EHP density *ρ* were described using the common expression for the dielectric function with the inter-band and intra-band (Drude) contributions (the first and second terms, respectively) [59]

_{pl}for the electron charge

*e*, optical mass of

*e-h*pairs

*m*

_{opt}

^{* = }

*m*

_{e}*

*m*

_{h}*/(

*m*

_{e}*

*+m*

_{h}*) ≈

*m*

_{e}* ∼

*m*

_{e}(for effective

*e,h*-masses

*m*

_{e}* «

*m*

_{h}*), and universal dielectric constant ε

_{0}reads

The electronic high-frequency dielectric constant ε_{HF} ≈ ε_{IB}, and e–h pair relaxation time in the Fermi-liquid approximation adapted from [60] is taken in the form

The electronic bandgap renormalization (shrinkage, ω* ≥ ω) [60] and band filling effect (ρ ≤ ρ_{bf}) [61] were neglected. The calculated magnitudes of ε* were converted into plasma density-dependent prompt true absorption and reflection coefficients, α_{0} and *R*, respectively, assuming Fresnel reflection from a homogeneous photo-excited surface layer of fluorite to yield the corresponding effective absorption coefficients α(ρ) = (1 − *R*(ρ))α_{0}(ρ).

The calculated effective absorption coefficients α(ρ) rapidly increased with respect to *ρ* (Fig. 6) for near-critical EHP densities (according to Eq. (14), *ρ*_{crit}(515 nm) ≈ 8 × 10^{21} cm^{-3} in CaF_{2}), saturating for *ρ* > *ρ*_{crit} (515 nm) at the level ≈ 1.2 × 10^{4} cm^{-1}, which is reasonably close to the experimental value α (515 nm) ≈ (1.8 ± 0.4) × 10^{4} cm^{-1}. Such self-consistent saturated effective absorption coefficient of the near- and supercritical EHP results from the strongly increasing true absorption coefficient α_{0} and reflection coefficient *R* (Fig. 6).

#### 3.4 Characteristic electron dynamics and related absorption mechanisms

Our experimental determination of narrow-crater size dependence on incident pulse energy enabled the evaluation of the optical nonlinearity, describing the femtosecond-laser energy deposition rate η in fluorite to facilitate ablation. This is generally not possible for broad, multi-micron craters. Below, we will discuss a few main regimes of femtosecond-laser generation EHP evolution and energy deposition dynamics, considering the standard rate equation [53] to provide useful insights into these dynamics.

The common rate equation for EHP (not multiple-rate equations, MRE [62]), not subjected to self-trapping inside the bandgap [63], can be described in the simplified form as follows

Here, the first term describes the saturable N-photon absorption/ionization (MPA(I), saturation threshold *ρ*_{0}); the second term represents the avalanche ionization (AI) transforming into free-carrier absorption (FCA) at sub/near-critical EHP densities ρ, and the third term indicates the reverse Auger recombination (AR) rate with its coefficient γ. Since the quantitative values of the material parameters *ρ*_{0} and γ are not known, as for many other dielectrics, we will next perform a qualitative analysis. For the given dielectric bandgap, it is based on incident femtosecond-laser intensity magnitudes and characteristic ranges of EHP density, which related to the critical density ρ_{crit} *at the given laser wavelength* and the characteristic Auger-recombination density ρ_{auger}; this makes the Auger-relaxation time ∼1/(γρ_{auger}^{2}) comparable to the laser pulse width.

Specifically, we can find several characteristic cases.

- ii) Medium NIR-laser intensities ∼ 10 TW/cm
^{2}, ρ ≤ ρ_{crit}< ρ_{auger}(unlimited MPI-seeded FCA): - iv) Low visible/UV laser intensities « 10 TW/cm
^{2}, ρ_{auger}< ρ < ρ_{crit}(Auger recombination-limited MPI):

In this study, particularly for the narrow craters in Fig. 3, we observed the slope ≈2 at 1030-nm pumping and ≈5 at 515-nm pumping, respectively, describing the femtosecond-laser energy deposition relationships with intensity during the ablation. At 515-nm pumping, the 5-photon dependence represents simply either case i) (ρ < ρ_{auger}, ρ_{crit}), or case iv) (ρ_{auger} < ρ < ρ_{crit}), describing the sub-critical EHP generation regime via the multi-photon absorption. In contrast, at 1030-nm pumping, the square dependence represents the interplay between the low-intensity MPI-dominated regimes i) and iv) and high-intensity FCA-dominated regime v) (ρ_{auger} < ρ < ρ_{crit}), which are temporally convoluted over the laser pulse.

## 4. Conclusions

In conclusion, single-shot femtosecond-laser surface ablation studies at both 1030- and 515-nm wavelengths revealed a double-crater structure, where the external shallow crater demonstrates its characteristic radius, considerably smaller than the focal radius, while the central deep crater follows the focal radius in its extension versus increasing femtosecond-laser pulse energy. The ablation depth varies differently in the corresponding intensity ranges, indicating potentially not only different ablation mechanisms – low-intensity spallation and high-intensity phase explosion, but also the corresponding multi-photon and free-carrier absorption mechanisms. For smaller intensities, the different size dependence for the shallow craters was related to their 3D energy deposition (absorption + transport) character, as compared to the broader high-intensity craters with their strong FCA absorption, resulting in 1D energy deposition. These finding are qualitatively justified by our spectral broadening studies for the 1030-nm and 515-nm pump pulses, enabling also to evaluate their wavelength-dependent Kerr coefficients. Our analysis of ultrafast laser energy deposition and electron dynamics in fluorite photo-excited by tightly focused near-IR and visible-range femtosecond laser pulses in the framework of the general rate equation supports the observed trends in terms of incident intensity, laser wavelength, and characteristic plasma density ranges.

## Funding

Ministry of Science and Higher Education of the Russian Federation (0705-2020-0041).

## Disclosures

The authors declare no conflicts of interest.

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